investigation of quasi-periodic order in geometrical structure of patkaneh

the quasi-periodic order is a type of geometrical order in universe which is observed in the structure of quasi-crystals. this order was first introduced in mathematics in 1973 by roger penrose and then discovered in quasi-crystals in 1984 while spreading in mathematics, chemistry and physics. the paper professor james lu et al published in science magazine in 2007 illustrated that iranians were familiar with the decagonal quasi-periodic order and used it in the structure of shah gireh from teymourid period on. in this paper, we aim to prove that iranian architects were familiar with both decagonal quasi-periodic order and octagonal quasi-periodic order. patkaneh is an element emerged from history and culture of iran-islam and is the product of iranian squinch development. the researchers such as besenval have related the application of squinch in sarvestan palace as the mother of all chapires (transitional region between square room and dome) in islamic architecture. gradually, repeating squinch on top of each other in different buildings brought about patkaneh and outstanding samples were formed. by patkaneh, we mean a type of muqarnas which has the self-static property (yet there are some decoration samples) and its geometry is based on a plan of square or rhombus of 45°. it seems that the common style of muqarnas to ilkhani period is what we name patkaneh,  following pirnia. to prove the existence of quasi-periodic order in the structure of patkaneh, we investigate the case in three levels. the first contains the similar repeated units with the octagonal quasi-periodic order structure. the second is related to the quasi-periodic order in central core and the third is pertained to the development based on principle of quasi-periodic order. the studies show that patkaneh has an octagonal quasi-periodic order in its central core and has capability to grow based on quasi-periodic order. despite existence of these two features, the studies of the third level of similarities make us conclude that; first, most patkanehs have four-axial symmetry in their developed structure which is in contradiction with the principles of quasi-periodic order, second, the existence of hybrid elements in the developed structure of patkaneh is as ignoring the principles of quasi-periodic order. totally, patkanehs do not follow the principles of quasi-periodic order in their development. therefore, the findings are: the first is the proof of quasi-periodic order in the structure of central core of patkaneh. it is claimed that iranians were familiar with the principles of octagonal quasi-periodic order which extends the claim of professor lu to ilkhanid era. the second is illustrating the capability of patkaneh in the change of the octagonal symmetry of central core to the four-axial symmetry in the surrounding context. this combines  two geometrical systems in iranian architecture. the first system is a geometrical system based on subscale structure of nature appearing as octagonal central core of patkaneh. the second system is a functional geometric system in architectural plan causing the space geometries to become square or rectangle. therefore, the geometry dominating patkaneh has the capabilities to combine these two systems.